In this paper we consider which families of finite simple groups
G
G
have the property that for each
ϵ
>
0
\epsilon > 0
there exists
N
>
0
N > 0
such that, if
|
G
|
≥
N
|G| \ge N
and
S
,
T
S, T
are normal subsets of
G
G
with at least
ϵ
|
G
|
\epsilon |G|
elements each, then every non-trivial element of
G
G
is the product of an element of
S
S
and an element of
T
T
.
We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form
P
S
L
n
(
q
)
\mathrm {PSL}_n(q)
where
q
q
is fixed and
n
→
∞
n\to \infty
. However, in the case
S
=
T
S=T
and
G
G
alternating this holds with an explicit bound on
N
N
in terms of
ϵ
\epsilon
.
Related problems and applications are also discussed. In particular we show that, if
w
1
,
w
2
w_1, w_2
are non-trivial words,
G
G
is a finite simple group of Lie type of bounded rank, and for
g
∈
G
g \in G
,
P
w
1
(
G
)
,
w
2
(
G
)
(
g
)
P_{w_1(G),w_2(G)}(g)
denotes the probability that
g
1
g
2
=
g
g_1g_2 = g
where
g
i
∈
w
i
(
G
)
g_i \in w_i(G)
are chosen uniformly and independently, then, as
|
G
|
→
∞
|G| \to \infty
, the distribution
P
w
1
(
G
)
,
w
2
(
G
)
P_{w_1(G),w_2(G)}
tends to the uniform distribution on
G
G
with respect to the
L
∞
L^{\infty }
norm.